___________________________________________________
Introduced by David R. Koenig 
 The Official Handbook for the PRM Certification
 
Preface I Zvi Wiener  
I.A.1.1 Introduction I.A.1.2 Mathematical Expectations: Prices or Utilities? I.A.1.3 The Axiom of Independence of Choice I.A.1.4 Maximising Expected Utility
I.A.1.4.1 The Four Basic Axioms I.A.1.4.2 Introducing the Utility Function I.A.1.4.3 Risk Aversion (and Risk Tolerance) I.A.1.4.4 Certain Equivalence I.A.1.4.5 Summary
I.A.1.5 Encoding a Utility Function I.A.1.5.1 For an Individual I.A.1.5.2 For a Firm I.A.1.5.3 Ironing out Anomalies
I.A.1.6 The Mean–Variance Criterion I.A.1.6.1 The Criterion I.A.1.6.2 Estimating Risk Tolerance I.A.1.6.3 Applications of the Mean–Variance Criterion
I.A.1.7 Risk-Adjusted Performance Measures I.A.1.7.1 The Sharpe Ratio I.A.1.7.2 RAPMs in an Equilibrium Market
I.A.1.7.2.1 The Treynor Ratio and Jensen’s Alpha I.A.1.7.2.2 Application of the Treynor Ratio I.A.1.7.2.3 Application of Jensen’s Alpha
I.A.1.7.3 Generalising Sharpe Ratios I.A.1.7.3.1 The Generalised Sharpe Ratio I.A.1.7.3.2 The Adjusted Sharpe Ratio
I.A.1.7.4 Downside RAPMs I.A.1.7.4.1 RAROC I.A.1.7.4.2 Sortino Ratio, Omega Index and other Kappa indices
I.A.1.8 Summary  Appendix I.A.1.A: Terminology  Appendix I.A.1.B: Utility Functions
I.A.1.B.1 The Exponential Utility Function I.A.1.B.2 The Logarithmic Utility Function I.A.1.B.3 The Quadratic Utility Function I.A.1.B.4 The Power Utility Function
I.A.2 Portfolio Mathematics  Paul Glasserman
I.A.2.1 Means and Variances of Past Returns
2004 © The Professional Risk Managers’ International Association ii
I.A.2.2 Mean and Variance of Future Returns I.A.2.2.1 Single Asset I.A.2.2.2 Covariance and Correlation I.A.2.2.3 Mean and Variance of a Linear Combination I.A.2.2.4 Example: Portfolio Return I.A.2.2.5 Example: Portfolio Profit I.A.2.2.6 Example: Long and Short Positions I.A.2.2.7 Example: Correlation
I.A.2.3 Mean-Variance Tradeoffs I.A.2.3.1 Achievable Expected Returns I.A.2.3.2 Achievable Variance and Standard Deviation I.A.2.3.3 Achievable Combinations of Mean and Standard Deviation I.A.2.3.4 Efficient Frontier I.A.2.3.5 Utility Maximization I.A.2.3.6 Varying the Correlation Parameter
I.A.2.4 Multiple Assets I.A.2.4.1 Portfolio Mean and Variance I.A.2.4.2 Vector Matrix Notation I.A.2.4.3 Efficient Frontier
I.A.2.5 A Hedging Example I.A.2.5.1 Problem Formulation I.A.2.5.2 Gallon-for-Gallon Hedge I.A.2.5.3 Minimum-Variance Hedge I.A.2.5.4 Effectiveness of the Optimal Hedge I.A.2.5.5 Connection with Regression
I.A.2.6 Serial Correlation I.A.2.7 Normally Distributed Returns
I.A.2.7.1 The Distribution of Portfolio Returns I.A.2.7.2 Value-at-Risk I.A.2.7.3 Probability of Reaching a Target I.A.2.7.4 Probability of Beating a Benchmark
I.A.3 Capital Allocation Keith Cuthbertson, Dirk Nitzsche  
I.A.3.1 An Overview I.A.3.1.1 Portfolio Diversification I.A.3.1.2 Tastes and Preferences for Risk versus Return
I.A.3.2 Mean–Variance Criterion I.A.3.3 Efficient Frontier: Two Risky Assets
I.A.3.3.1 Different Values of the Correlation Coefficient I.A.3.4 Asset Allocation
I.A.3.4.1 The efficient frontier: n risky assets I.A.3.5 Combining the Risk-Free Asset with Risky Assets I.A.3.6 The Market Portfolio and the CML I.A.3.7 The Market Price of Risk and the Sharpe Ratio I.A.3.8 Separation Principle I.A.3.9 Summary
 Appendix: Mathematics of the Mean–Variance Model
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I.A.4 The CAPM and Multifactor Models Keith Cuthbertson, Dirk Nitzsche
I.A.4.1 Overview I.A.4.2 Capital Asset Pricing Model
I.A.4.2.1 Estimating Beta I.A.4.2.2 Beta and Systematic Risk
I.A.4.3 Security Market Line I.A.4.4 Performance Measures
I.A.4.4.1 Sharpe Ratio I.A.4.4.2 Jensen’s ‘alpha’
I.A.4.5 The Single-Index Model I.A.4.6 Multifactor Models and the APT
I.A.4.6.1 Portfolio Returns I.A.4.7 Summary
I.A.5 Basics of Capital Structure Steven Bishop
I.A.5.1 Introduction I.A.5.2 Maximising Shareholder Value, Incentives and Agency Costs
I.A.5.2.1 Agency Costs I.A.5.2.1.1 Agency Cost of Equity I.A.5.2.1.2 Agency Costs of Debt
I.A.5.2.2 Information Asymmetries I.A.5.3 Characteristics of Debt and Equity I.A.5.4 Choice of Capital Structure
I.A.5.4.1 Do not think debt is attractive because the interest rate is lower than the cost of equity!
I.A.5.4.2 Debt can be attractive I.A.5.4.2.1 Differential treatment of payments to debt-holders and
shareholders I.A.5.4.2.2 Greater Flexibility I.A.5.4.2.3 Monitoring ‘improves’ performance and reduces the
negative aspect of information asymmetry I.A.5.4.2.4 Debt enforces a discipline of paying out operating earnings I.A.5.4.2.5 Debt financing avoids negative signals about
management’s view of the value of equity I.A.5.4.3 Debt can also be unattractive
I.A.5.4.3.1 Exposure to bankruptcy costs I.A.5.4.3.2 Exposure to financial distress costs I.A.5.4.3.3 Agency costs
I.A.5.4.4 Thus choose the point where disadvantages offset advantages I.A.5.5 Making the capital structure decision
I.A.5.5.1 Guidelines I.A.5.5.2 What do CFOs say they consider when making a capital structure choice?
I.A.5.6 Conclusion
I.A.6 The Term Structure of Interest Rates Deborah Cernauskas, Elias Demetriades
I.A.6.1 Compounding Methods I.A.6.1.1 Continuous versus Discrete Compounding I.A.6.1.2 Annual Compounding versus More Regular Compounding I.A.6.1.3 Periodic Interest Rates versus Effective Annual Yield
I.A.6.2 Term Structure – A Definition
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I.A.6.3 Shapes of the Yield Curve I.A.6.4 Spot and Forward Rates I.A.6.5 Term Structure Theories
I.A.6.5.1 Pure or Unbiased Expectations I.A.6.5.2 Liquidity Preference I.A.6.5.3 Market Segmentation
I.A.6.6 Summary
I.A.7.1 The Difference between Pricing and Valuation for Forward Contracts I.A.7.2 Principles of Pricing and Valuation for Forward Contracts on Assets
I.A.7.2.1 The Value at Time 0 of a Forward Contract I.A.7.2.2 The Value at Expiration of a Forward Contract on an Asset I.A.7.2.3 The Value Prior to Expiration of a Forward Contract on an Asset I.A.7.2.4 The Value of a Forward Contract on an Asset when there are
Cash Flows on the Asset during the Life of the Contract I.A.7.2.5 Establishing the Price of a Forward Contract on an Asset I.A.7.2.6 Pricing and Valuation when the Cash Flows or Holding Costs are
Continuous I.A.7.2.7 Numerical Examples
I.A.7.3 Principles of Pricing and Valuation for Forward Contracts on Interest Rates I.A.7.3.1 The Value of an FRA at Expiration I.A.7.3.2 The Value of an FRA at the Start I.A.7.3.3 The Value of an FRA During Its Life I.A.7.3.4 Pricing the FRA on Day 0 I.A.7.3.5 Numerical Examples
I.A.7.4 The Relationship Between Forward and Futures Prices
I.A.8 Basic Principles of Option Pricing Paul Wilmott
I.A.8.1 Factors Affecting Option Prices I.A.8.2 Put–Call Parity I.A.8.3 One-step Binomial Model and the Riskless Portfolio I.A.8.4 Delta Neutrality and Simple Delta Hedging I.A.8.5 Risk-Neutral Valuation I.A.8.6 Real versus Risk-Neutral I.A.8.7 The Black–Scholes–Merton Pricing Formula I.A.8.8 The Greeks I.A.8.9 Implied Volatility I.A.8.10 Intrinsic versus Time Value
B – FINANCIAL INSTRUMENTS
I.B.1 General Characteristics of Bonds Lionel Martellini, Philippe Priaulet
I.B.1.1 Definition of a Bullet Bond I.B.1.2 Terminology and Convention I.B.1.3 Market Quotes
I.B.1.3.1 Bond Quoted Price I.B.1.3.2 Bond Quoted Yield I.B.1.3.3 Bond Quoted Spread I.B.1.3.4 Liquidity Spreads
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I.B.1.5 Summary
I.B.2 The Analysis of Bonds  Moorad Choudhry
I.B.2.1 Features of Bonds I.B.2.1.1 Type of Issuer I.B.2.1.2 Term to Maturity I.B.2.1.3 Principal and Coupon Rate I.B.2.1.4 Currency
I.B.2.2 Non-conventional Bonds I.B.2.2.1 Floating-Rate Notes I.B.2.2.2 Index-Linked Bonds I.B.2.2.3 Zero-Coupon Bonds I.B.2.2.4 Securitised Bonds I.B.2.2.5 Bonds with Embedded Options
I.B.2.3 Pricing a Conventional Bond I.B.2.3.1 Bond Cash Flows I.B.2.3.2 The Discount Rate I.B.2.3.3 Conventional Bond Pricing I.B.2.3.4 Pricing Undated Bonds I.B.2.3.5 Pricing Conventions I.B.2.3.6 Clean and Dirty Bond Prices: Accrued Interest
I.B.2.4 Market Yield I.B.2.4.1 Yield Measurement I.B.2.4.2 Current Yield I.B.2.4.3 Yield to Maturity
I.B.2.5 Relationship between Bond Yield and Bond Price I.B.2.6 Duration
I.B.2.6.1 Calculating Macaulay Duration and Modified Duration I.B.2.6.2 Properties of the Macaulay Duration I.B.2.6.3 Properties of the Modified Duration
I.B.2.7 Hedging Bond Positions I.B.2.8 Convexity I.B.2.9 A Summary of Risks Associated with Bonds
I.B.3 Futures and Forwards Keith Cuthbertson, Dirk Nitzsche
I.B.3.1 Introduction I.B.3.2 Stock Index Futures
I.B.3.2.1 Contract Specifications I.B.3.2.2 Index arbitrage and program trading I.B.3.2.3 Hedging Using Stock Index Futures I.B.3.2.4 Tailing the Hedge I.B.3.2.5 Summary
I.B.3.3 Currency Forwards and Futures I.B.3.3.1 Currency Forward Contracts I.B.3.3.2 Currency Futures Contracts I.B.3.3.3 Hedging Currency Futures and Forwards I.B.3.3.4 Summary
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I.B.3.6.1 US T-bill Futures I.B.3.6.2 Three-Month Eurodollar Futures I.B.3.6.3 Sterling Three-Month Futures I.B.3.6.4 Hedging Interest-Rate Futures I.B.3.6.5 Hedge Ratios I.B.3.6.6 Hedging Using US T-bill Futures I.B.3.6.7 Summary
I.B.3.7 T-bond Futures I.B.3.7.1 Contract Specifications
I.B.3.7.1.1 UK Long Gilt Futures Contract I.B.3.7.1.2 US T-bond Futures Contract
I.B.3.7.2 Conversion Factor and Cheapest to Deliver I.B.3.7.3 Hedging Using T-bond Futures I.B.3.7.4 Hedging a Single Bond I.B.3.7.5 Hedging a Portfolio of Bonds I.B.3.7.6 Summary
I.B.3.8 Stack and Strip Hedges I.B.3.9 Concluding Remarks
I.B.4 Swaps Salih Neftci
I.B.4.2.1 Equity Swaps I.B.4.2.2 Commodity Swaps I.B.4.2.3 Interest Rate Swaps I.B.4.2.4 Currency Swaps I.B.4.2.5 Basis Swaps I.B.4.2.6 Volatility Swaps
I.B.4.3 Engineering Interest-Rate Swaps I.B.4.4 Risk of Swaps
I.B.4.4.1 Market Risk I.B.4.4.2 Credit Risk and Counterparty Risk I.B.4.4.3 Volatility and Correlation Risk
I.B.4.5 Other Swaps I.B.4.6 Uses of Swaps
I.B.4.6.1 Uses of Equity Swaps I.B.4.7 Swap Conventions
I.B.5 Vanilla Options Paul Wilmott
I.B.5.1 Stock Options – Characteristics and Payoff Diagrams I.B.5.2 American versus European Options I.B.5.3 Strategies Involving a Single Option and a Stock I.B.5.4 Spread Strategies
I.B.5.4.1 Bull and Bear Spreads I.B.5.4.2 Calendar Spreads
I.B.5.5 Other Strategies I.B.5.5.1 Straddles and Strangles I.B.5.5.2 Risk Reversal
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I.B.6 Credit Derivatives  Moorad Choudhry
I.B.6.1 Introduction I.B.6.1.1 Why Use Credit Derivatives? I.B.6.1.2 Classification of Credit Derivative Instruments I.B.6.1.3 Definition of a Credit Event
I.B.6.2 Credit Default Swaps I.B.6.3 Credit-Linked Notes I.B.6.4 Total Return Swaps
I.B.6.4.1 Synthetic Repo I.B.6.4.2 Reduction in Credit Risk I.B.6.4.3 Capital Structure Arbitrage I.B.6.4.4 The TRS as a Funding Instrument
I.B.6.5 Credit Options I.B.6.6 Synthetic Collateralised Debt Obligations
I.B.6.6.1 Cash Flow CDOs I.B.6.6.2 What is a Synthetic CDO? I.B.6.6.3 Funding Synthetic CDOs I.B.6.6.4 Variations in Synthetic CDOs I.B.6.6.5 Use of Synthetic CDOs I.B.6.6.6 Advantages and Limitations of Synthetic Structures
I.B.6.7 General Applications of Credit Derivatives I.B.6.7.1 Use of Credit Derivatives by Portfolio Managers
I.B.6.7.1.1 Enhancing portfolio returns I.B.6.7.1.2 Reducing credit exposure I.B.6.7.1.3 Credit switches and zero-cost credit exposure I.B.6.7.1.4 Exposure to market sectors I.B.6.7.1.5 Trading Credit spreads
I.B.6.7.2 Use of Credit Derivatives by Banks I.B.6.8 Unintended Risks in Credit Derivatives I.B.6.9 Summary
I.B.7 Caps, Floors & Swaptions Lionel Martellini, Philippe Priaulet
I.B.7.1 Caps, Floors and Collars: Definition and Terminology I.B.7.2 Pricing Caps, Floors and Collars
I.B.7.2.1 Cap Formula I.B.7.2.2 Floor Formula I.B.7.2.3 Market Quotes
I.B.7.3 Uses of Caps, Floors and Collars I.B.7.3.1 Limiting the Financial Cost of Floating-Rate Liabilities I.B.7.3.2 Protecting the Rate of Return of a Floating-Rate Asset
I.B.7.4 Swaptions: Definition and Terminology I.B.7.5 Pricing Swaptions
I.B.7.5.1 European Swaption Pricing Formula I.B.7.5.2 Market Quotes
I.B.7.6 Uses of Swaptions I.B.7.7 Summary
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I.B.8.1 Introduction I.B.8.1.1 Convertibles – a definition I.B.8.1.2 Convertible Bond Market Size I.B.8.1.3 A Brief History
I.B.8.2 Characteristics of Convertibles I.B.8.2.1 Relationship with Stock Price I.B.8.2.2 Call and Put Features I.B.8.2.3 Players in the Convertible Bond Market I.B.8.2.4 Convertible Bond Funds I.B.8.2.5 Convertible Arbitrage Hedge Funds
I.B.8.3 Capital Structure Implications (for Banks) I.B.8.4 Mandatory Convertibles I.B.8.5 Valuation and Risk Assessment I.B.8.6 Summary
I.B.9 Simple Exotics Catriona March
I.B.9.1 Introduction I.B.9.2 A Short History I.B.9.3 Classifying Exotics I.B.9.4 Notation I.B.9.5 Digital Options
I.B.9.5.1 Cash-or-Nothing Options I.B.9.5.2 Asset-or-Nothing Options I.B.9.5.3 Vanillas and Digitals as Building Blocks I.B.9.5.4 Contingent Premium Options I.B.9.5.5 Range Notes I.B.9.5.6 Managing Digital Options
I.B.9.6 Two Asset Options I.B.9.6.1 Product and Quotient Options I.B.9.6.2 Exchange Options I.B.9.6.3 Outperformance Options I.B.9.6.4 Other Two-Colour Rainbow Options I.B.9.6.5 Spread Options I.B.9.6.6 Correlation Risk
I.B.9.7 Quantos I.B.9.7.1 Foreign Asset Option Struck in Foreign Currency I.B.9.7.2 Foreign Asset Option Struck in Domestic Currency I.B.9.7.3 Implied Correlation I.B.9.7.4 Foreign Asset Linked Currency Option I.B.9.7.5 Guaranteed Exchange Rate Foreign Asset Options
I.B.9.8 Second-Order Contracts I.B.9.8.1 Compound Options I.B.9.8.2 Typical Uses of Compound Options I.B.9.8.3 Instalment Options I.B.9.8.4 Extendible Options
I.B.9.9 Decision Options I.B.9.9.1 American Options I.B.9.9.2 Bermudan Options I.B.9.9.3 Shout Options
I.B.9.10 Average Options I.B.9.10.1 Average Rate and Average Strike Options
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I.B.9.11 Options on Baskets of Assets I.B.9.11.1 Basket Options I.B.9.11.2 Pricing and Hedging Basket Options I.B.9.11.3 Mountain Options
I.B.9.12 Barrier and Related Options I.B.9.12.1 Single-Barrier Options I.B.9.12.2 No-Touch, One-Touch and Rebates I.B.9.12.3 Partial-Barrier Options I.B.9.12.4 Double-Barrier Options I.B.9.12.5 Even More Barrier Options I.B.9.12.6 Relationships I.B.9.12.7 Ladders I.B.9.12.8 Lookback and Hindsight Options
I.B.9.13 Other Path-Dependent Options I.B.9.13.1 Forward Start Options I.B.9.13.2 Reset Options I.B.9.13.3 Cliquet Options
I.B.9.14 Resolution Methods I.B.9.15 Summary
C - MARKETS
I.C.1 The Structure of Financial Markets Colin Lawrence, Alistair Milne  
I.C.1.1 Introduction I.C.1.2 Global Markets and Their Terminology I.C.1.3 Drivers of Liquidity
I.C.1.3.1 Repo Markets I.C.1.4 Liquidity and Financial Risk Management I.C.1.5 Exchanges versus OTC Markets I.C.1.6 Technological Change I.C.1.7 Post-trade Processing I.C.1.8 Retail and Wholesale Brokerage I.C.1.9 New Financial Markets I.C.1.10 Conclusion
I.C.2 The Money Markets Canadian Securities Institute
I.C.2.1 Introduction I.C.2.2 Characteristics of Money Market Instruments I.C.2.3 Deposits and Loans
I.C.2.3.1 Deposits from Businesses I.C.2.3.2 Loans to Businesses I.C.2.3.3 Repurchase Agreements I.C.2.3.4 International Markets I.C.2.3.5 The London Interbank Offered Rate (LIBOR)
I.C.2.4 Money Market Securities I.C.2.4.1 Treasury Bills I.C.2.4.2 Commercial Paper I.C.2.4.3 Bankers’ Acceptances I.C.2.4.4 Certificates of Deposit
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I.C.3 The Bond Market  Moorad Choudhry, Lionel Martellini, Philippe Priaulet
I.C.3.1 Introduction I.C.3.2 The Players
I.C.3.2.1 Intermediaries and Banks I.C.3.2.2 Institutional Investors I.C.3.2.3 Market Professionals
I.C.3.3 Bonds by Issuers I.C.3.3.1 Government Bonds I.C.3.3.2 US Agency Bonds I.C.3.3.3 Municipal Bonds I.C.3.3.4 Corporate Bonds I.C.3.3.5 Eurobonds (International Bonds)
I.C.3.4 The Markets I.C.3.4.1 The Government Bond Market I.C.3.4.2 The Corporate Bond Market
I.C.3.4.2.1 The market by country and sector I.C.3.4.2.2 Underwriting a new issue
I.C.3.4.3 The Eurobond Market I.C.3.4.4 Market Conventions
I.C.3.5 Credit Risk I.C.3.6 Summary
I.C.4 The Foreign Exchange Market Canadian Securities Institute, Toronto
I.C.4.1 Introduction I.C.4.2 The Interbank Market I.C.4.3 Exchange-Rate Quotations
I.C.4.3.1 Direct Dealing I.C.4.3.2 Foreign Exchange Brokers I.C.4.3.3 Electronic Brokering Systems I.C.4.3.4 The Role of the US Dollar I.C.4.3.5 Market and Quoting Conventions I.C.4.3.6 Cross Trades and Cross Rates
I.C.4.4 Determinants of Foreign Exchange Rates I.C.4.4.1 The Fundamental Approach I.C.4.4.2 A Short-Term Approach I.C.4.4.3 Central Bank Intervention
I.C.4.5 Spot and Forward Markets I.C.4.5.1 The Spot Market I.C.4.5.2 The Forward Market
I.C.4.5.2.1 Forward Discounts and Premiums I.C.4.5.2.2 Interest-Rate Parity
I.C.4.6 Structure of a Foreign Exchange Operation I.C.4.7 Summary/Conclusion
I.C.5 The Stock Market  Andrew Street
I.C.5.1 Introduction I.C.5.2 The Characteristics of Common Stock
I.C.5.2.1 Share Premium and Capital Accounts and Limited Liability
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I.C.5.2.2 Equity Shareholder’s Rights and Dividends I.C.5.2.3 Other Types of Equity Shares – Preference Shares I.C.5.2.4 Equity Price Data I.C.5.2.5 Market Capitalisation (or ‘Market Cap’) I.C.5.2.6 Stock Market Indices I.C.5.2.7 Equity Valuation
I.C.5.3 Stock Markets and their Participants I.C.5.3.1 The Main Participants – Firms, Investment Banks and Investors I.C.5.3.2 Market Mechanics
I.C.5.4 The Primary Market – IPOs and Private Placements I.C.5.4.1 Basic Primary Market Process I.C.5.4.2 Initial Public Offerings I.C.5.4.3 Private Placements
I.C.5.5 The Secondary Market – the Exchange versus OTC Market I.C.5.5.1 The Exchange I.C.5.5.2 The Over-the-Counter Market I.C.5.6 Trading Costs I.C.5.6.1 Commissions I.C.5.6.2 Bid–Offer Spread I.C.5.6.3 Market Impact
I.C.5.7 Buying on Margin I.C.5.7.1 Leverage I.C.5.7.2 Percentage Margin and Maintenance Margin I.C.5.7.3 Why Trade on Margin?
I.C.5.8 Short Sales and Stock Borrowing Costs I.C.5.8.1 Short Sale I.C.5.8.2 Stock Borrowing
I.C.5.9 Exchange-Traded Derivatives on Stocks I.C.5.9.1 Single Stock and Index Options I.C.5.9.2 Expiration Dates I.C.5.9.3 Strike Prices I.C.5.9.4 Flex Options I.C.5.9.5 Dividends and Corporate Actions I.C.5.9.6 Position Limits I.C.5.9.7 Trading I.C.5.10 Summary
I.C.6 The Futures Market Canadian Securities Institute
I.C.6.1 Introduction I.C.6.2 History of Forward-Based Derivatives and Futures Markets I.C.6.3 Futures Contracts and Markets
I.C.6.3.1 General Characteristics of Futures Contracts and Markets I.C.6.3.2 Settlement of Futures Contracts I.C.6.3.3 Types of Orders I.C.6.3.4 Margin Requirements and Marking to Market I.C.6.3.5 Leverage I.C.6.3.6 Reading a Futures Quotation Page I.C.6.3.7 Liquidity and Trading Costs
I.C.6.4 Options on Futures I.C.6.5 Futures Exchanges and Clearing Houses
I.C.6.5.1 Exchanges I.C.6.5.2 Futures Exchange Functions I.C.6.5.3 Clearing Houses I.C.6.5.4 Marking-to-Market and Margin
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I.C.6.7.1 Locals I.C.6.7.2 Day Traders I.C.6.7.3 Position Traders I.C.6.7.4 Spreaders
I.C.6.7.4.1 Intramarket Spreads I.C.6.7.4.2 Intercommodity Spreads I.C.6.7.4.3 Intermarket Spreads I.C.6.7.4.4 Commodity Product Spread
I.C.6.8 Market Participants – Managed Futures Investors I.C.6.9 Summary and Conclusion
I.C.7 The Structure of Commodities Markets  Colin Lawrence, Alistair Milne  
I.C.7.1 Introduction I.C.7.2 The Commodity Universe and Anatomy of Markets
I.C.7.2.1 Commodity Types and Characteristics I.C.7.2.2 The Markets for Trading I.C.7.2.3 Delivery and Settlement Methods I.C.7.2.4 Commodity Market Liquidity I.C.7.2.5 The Special Case of Gold as a Reserve Asset
I.C.7.3 Spot–Forward Pricing Relationships I.C.7.3.1 Backwardation and Contango I.C.7.3.2 Reasons for Backwardation I.C.7.3.3 The No-Arbitrage Condition
I.C.7.4 Short Squeezes, Corners and Regulation I.C.7.4.1 Historical Experience I.C.7.4.2 The Exchange Limits
I.C.7.5 Risk Management at the Commodity Trading Desk I.C.7.6 The Distribution of Commodity Returns
I.C.7.6.1 Evidence of Non-normality I.C.7.6.2 What Drives Commodity Prices?
I.C.7.7 Conclusions
I.C.8.1 Introduction I.C.8.2 Market Overview
I.C.8.2.1 The Products I.C.8.2.2 The Risks I.C.8.2.3 Developing a Cash Market
I.C.8.3 Energy Futures Markets I.C.8.3.1 The Exchanges I.C.8.3.2 The Contracts I.C.8.3.3 Options on Energy Futures I.C.8.3.4 Hedging in Energy Futures Markets I.C.8.3.5 Physical Delivery I.C.8.3.6 Market Changes: Backwardation and Contango
I.C.8.4 OTC Energy Derivative Markets I.C.8.4.1 The Singapore Market I.C.8.4.2 The European Energy Markets I.C.8.4.3 The North American Markets
I.C.8.5 Emerging Energy Commodity Markets
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I.C.8.6 The Future of Energy Trading I.C.8.6.1 Re-emergence of Speculative Trading? I.C.8.6.2 Electronic Energy Trading I.C.8.6.3 Trading in Asian Markets
I.C.8.7 Conclusion
Preface II  Carol Alexander
II.A.1 Symbols and Rules II.A.1.1 Expressions, Functions, Graphs, Equations and Greek II.A.1.2 The Algebra of Number II.A.1.3 The Order of Operations
II.A.2 Sequences and Series II.A.2.1 Sequences II.A.2.2 Series
II.A.3 Exponents and Logarithms II.A.3.1 Exponents II.A.3.2 Logarithms II.A.3.3 The Exponential Function and Natural Logarithms
II.A.4 Equations and Inequalities II.A.4.1 Linear Equations in One Unknown II.A.4.2 Inequalities II.A.4.3 Systems of Linear Equations in More Than One Unknown II.A.4.4 Quadratic Equations
II.A.5 Functions and Graphs II.A.5.1 Functions II.A.5.2 Graphs II.A.5.3 The Graphs of Some Functions
II.A.6 Case Study − Continuous Compounding II.A.6.1 Repeated Compounding II.A.6.2 Discrete versus Continuous Compounding
II.A.7 Summary
II.B.1 Introduction II.B.2 Data
II.B.2.1 Continuous and Discrete Data II.B.2.2 Grouped Data II.B.2.3 Graphical Representation of Data
II.B.2.3.1 The Frequency Bar Chart II.B.2.3.2 The Relative Frequency Distribution II.B.2.3.3 The Cumulative Frequency Distribution II.B.2.3.4 The Histogram
II.B.3 The Moments of a Distribution
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II.B.4 Measures of Location or Central Tendency – Averages II.B.4.1 The Arithmetic Mean II.B.4.2 The Geometric Mean II.B.4.3 The Median and the Mode
II.B.5 Measures of Dispersion II.B.5.1 Variance II.B.5.2 Standard Deviation II.B.5.3 Case Study: Calculating Historical Volatility from Returns Data II.B.5.4 The Negative Semi-variance and Negative Semi-deviation II.B.5.5 Skewness II.B.5.6 Kurtosis
II.B.6 Bivariate Data II.B.6.1 Covariance II.B.6.2 The Covariance Matrix II.B.6.3 The Correlation Coefficient II.B.6.4 The Correlation Matrix II.B.6.5 Case Study: Calculating the Volatility of a Portfolio
II.C Calculus Keith Parramore, Terry Watsham
II.C.1 Differential Calculus II.C.1.1 Functions II.C.1.2 The First Derivative II.C.1.3 Notation II.C.1.4 Simple Rules
II.C.1.4.1 Differentiating Constants II.C.1.4.2 Differentiating a Linear Function II.C.1.4.3 The Gradient of a Straight Line II.C.1.4.4 The Derivative of a Power of x II.C.1.4.5 Differentiating a scalar multiple of a function II.C.1.4.6 Differentiating the Sum of Two Functions of x II.C.1.4.7 Differentiating the Product of Two Functions of x II.C.1.4.8 Differentiating the Quotient of Two Functions of x II.C.1.4.9 Differentiating a Function of a Function II.C.1.4.10 Differentiating the Exponential Function II.C.1.4.11 Differentiating the Natural Logarithmic Function
II.C.2 Case Study: Modified Duration of a Bond II.C.3 Higher-Order Derivatives
II.C.3.1 Second Derivatives II.C.3.2 Further Derivatives II.C.3.3 Taylor Approximations
II.C.4 Financial Applications of Second Derivatives II.C.4.1 Convexity II.C.4.2 Convexity in Action II.C.4.3 The Delta and Gamma of an Option
II.C.5 Differentiating a Function of More than One Variable II.C.5.1 Partial Differentiation II.C.5.2 Total differentiation
II.C.6 Integral Calculus II.C.6.1 Indefinite and Definite Integrals II.C.6.2 Rules for Integration II.C.6.3 Guessing
II.C.7 Optimisation II.C.7.1 Finding the Minimum or Maximum of a Function of One Variable II.C.7.2 Maxima and Minima of Functions of More than One Variable
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II.D Linear Algebra Keith Parramore, Terry Watsham
II.D.1 Matrix Algebra II.D.1.1 Matrices II.D.1.2 Vectors and Transposes II.D.1.3 Manipulation of Matrices II.D.1.4 Matrix Multiplication II.D.1.5 Inverting a Matrix
II.D.2 Application of Matrix Algebra to Portfolio Construction II.D.2.1 Calculating the Risk of an Existing Portfolio II.D.2.2 Deriving Asset Weights for the Minimum Risk Portfolio II.D.2.3 Hedging a Vanilla Option Position
II.D.2.3.1 Calculating the position delta II.D.2.3.2 Establishing the delta-neutral hedge II.D.2.3.3 Gamma neutrality II.D.2.3.4 Vega neutrality II.D.2.3.5 Hedging a short option position
II.D.3 Quadratic Forms II.D.3.1 The Variance of Portfolio Returns as a Quadratic Form II.D.3.2 Definition of Positive Definiteness
II.D.4 Cholesky Decomposition II.D.4.1 The Cholesky Arithmetic II.D.4.2 Simulation in Excel
II.D.5 Eigenvalues and Eigenvectors II.D.5.1 Matrices as Transformations II.D.5.2 Definition of Eigenvector and Eigenvalue II.D.5.3 Determinants II.D.5.4 The Characteristic Equation
II.D.5.4.1 Testing for Positive Semi-definiteness II.D.5.4.2 Using the characteristic equation to find the eigenvalues of a covariance matrix II.D.5.4.3 Eigenvalues and eigenvectors of covariance and correlation matrices
II.D.5.5 Principal Components
II.E.1 Definitions and Rules II.E.1.1 Definitions
II.E.1.1.1 The classical approach II.E.1.1.2 The Bayesian approach
II.E.1.2 Rules for Probability II.E.1.2.1 (A or B) and (A and B) II.E.1.2.2 Conditional Probability
II.E.2 Probability Distributions II.E.2.1 Random Variables
II.E.2.1.1 Discrete Random Variables II.E.2.1.2 Continuous Random Variables
II.E.2.2 Probability Density Functions and Histograms II.E.2.3 The Cumulative Distribution Function II.E.2.4 The Algebra of Random Variables
II.E.2.4.1 Scalar Multiplication of a Random Variable
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II.E.2.5 The Expected Value of a Discrete Random Variable II.E.2.6 The Variance of a Discrete Random Variable II.E.2.7 The Algebra of Continuous Random Variables
II.E.3 Joint Distributions II.E.3.1 Bivariate Random Variables II.E.3.2 Covariance II.E.3.3 Correlation II.E.3.4 The Expected Value and Variance of a Linear Combination of Random
 Variables II.E.4 Specific Probability Distributions
II.E.4.1 The Binomial Distribution II.E.4.1.1 Calculating the ‘Number of Ways’ II.E.4.1.2 Calculating the Probability of r Successes II.E.4.1.3 Expectation and Variance
II.E.4.2 The Poisson Distribution II.E.4.2.1 Illustrations II.E.4.2.2 Expectation and Variance
II.E.4.3 The Uniform Continuous Distribution II.E.4.4.1 Normal Curves II.E.4.4.2 The Standard Normal Probability Density Function II.E.4.4.3 Finding Areas under a Normal Curve Using Excel
II.E.4.5 The Lognormal Probability Distribution II.E.4.5.1 Lognormal Curves II.E.4.5.2 The Lognormal Distribution Applied to Asset Prices II.E.4.5.3 The Mean and Variance of the Lognormal Distribution II.E.4.5.4 Application of the Lognormal Distribution to Future Asset Prices [ not in PRM exam]  
II.E.4.6 Student’s t Distribution II.E.4.7 The Bivariate Normal Distribution
II.F Regression  Keith Parramore, Terry Watsham
II.F.1 Simple Linear Regression II.F.1.1 The Model II.F.1.2 The Scatter Plot II.F.1.3 Estimating the Parameters
II.F.2 Multiple Linear Regression II.F.2.1 The model II.F.2.2 Estimating the Parameters
II.F.3 Evaluating the Regression Model II.F.3.1 Intuitive Interpretation II.F.3.2 Adjusted R2 II.F.3.3 Testing for Statistical Significance
II.F.4 Confidence Intervals II.F.4.1 Confidence Intervals for the Regression Parameters
II.F.5 Hypothesis Testing II.F.5.1 Significance Tests for the Regression Parameters II.F.5.2 Significance Test for R2 II.F.5.3 Type I and type II errors
II.F.6 Prediction II.F.7 Breakdown of the OLS Assumptions
II.F.7.1 Heteroscedasticity II.F.7.2 Autocorrelation II.F.7.3 Multicollinearity
II.F.8 Random Walks and Mean-Reversion
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II.G Numerical Methods Keith Parramore, Terry Watsham
II.G.1 Solving (Non-differential) Equations II.G.1.1 Three Problems II.G.1.2 Bisection II.G.1.3 Newton–Raphson II.G.1.4 Goal Seek
II.G.2 Numerical Optimisation II.G.2.1 The Problem II.G.2.2 Unconstrained Numerical Optimisation II.G.2.3 Constrained Numerical Optimisation II.G.2.4 Portfolio Optimisation Revisited
II.G.3 Numerical Methods for Valuing Options II.G.3.1 Binomial Lattices II.G.3.2 Finite Difference Methods II.G.3.3 Simulation
II.G.4 Summary
Preface III  Elizabeth Sheedy  
III.0 Capital Allocation and Risk Adjusted Performance  Andrew Aziz, Dan Rosen
III.0.1 Introduction III.0.1.1 Role of Capital in Financial Institution III.0.1.2 Types of Capital III.0.1.3 Capital as a Management Tool
III.0.2 Economic Capital III.0.2.1 Understanding Economic Capital III.0.2.2 The Top-Down Approach to Calculating Economic Capital
III.0.2.2.1 Top-Down Earnings Volatility Approach III.0.2.2.2 Top-Down Option-Theoretic Approach
III.0.2.3 The Bottom-Up Approach to Calculating Economic Capital III.0.2.4 Stress Testing of Portfolio Losses and Economic Capital III.0.2.5 Enterprise Capital Practices – Aggregation III.0.2.6 Economic Capital as Insurance for the Value of the Firm
III.0.3 Regulatory Capital III.0.3.1 Regulatory Capital Principles III.0.3.2 The Basel Committee of Banking Supervision and the Basel Accord III.0.3.3 Basel I Regulation
III.0.3.3.1 Minimum Capital Requirements under Basel I III.0.3.3.2 Regulatory Arbitrage under Basel I III.0.3.3.3 Meeting Capital Adequacy Requirements
III.0.3.4 Basel II Accord – Latest Proposals III.0.3.4.1 Pillar 1 - Minimum Capital Requirements III.0.3.4.2 Pillar 2 - Supervisory Review III.0.3.4.3 Pillar 3 - Market Discipline
III.0.3.5 A Simple Derivation of Regulatory Capital III.0.4 Capital Allocation and Risk Contributions
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III.0.4.2.1 Stand-alone EC Contributions III.0.4.2.2 Incremental EC Contributions III.0.4.2.3 Marginal EC Contributions III.0.4.2.4 Alternative Methods for Additive Contributions
III.0.5 RAROC and Risk-Adjusted Performance III.0.5.1 Objectives of RAPM III.0.5.2 Mechanics of RAROC III.0.5.3 RAROC and Capital Allocation Methodologies
III.0.6 Summary and Conclusions
III.A.1.1 Introduction III.A.1.2 Market Risk
III.A.1.2.1 Why is Market Risk Management Important? III.A.1.2.2 Distinguishing Market Risk from Other Risks
III.A.1.3 Market Risk Management Tasks III.A.1.4 The Organisation of Market Risk Management III.A.1.5 Market Risk Management in Fund Management
III.A.1.5.1 Market Risk in Fund Management III.A.1.5.2 Identification III.A.1.5.3 Assessment III.A.1.5.4 Control/Mitigation
III.A.1.6 Market Risk Management in Banking III.A.1.6.1 Market Risk in Banking III.A.1.6.2 Identification III.A.1.6.3 Assessment III.A.1.6.4 Control/Mitigation
III.A.1.7 Market Risk Management in Non-financial Firms III.A.1.7.1 Market Risk in Non-Financial Firms III.A.1.7.2 Identification III.A.1.7.3 Assessment III.A.1.7.4 Control/Mitigation
III.A.1.8 Summary
III.A.2 Introduction to Value at Risk Models  Kevin Dowd, David Rowe
III.A.2.1 Introduction III.A.2.2 Definition of VaR III.A.2.3 Internal Models for Market Risk Capital III.A.2.4 Analytical VaR Models III.A.2.5 Monte Carlo Simulation VaR
III.A.2.5.1 Methodology III.A.2.5.2 Applications of Monte Carlo simulation III.A.2.5.3 Advantages and Disadvantages of Monte Carlo VaR
III.A.2.6 Historical Simulation VaR III.A.2.6.1 The Basic Method III.A.2.6.2 Weighted historical simulation III.A.2.6.3 Advantages and Disadvantages of Historical Approaches
III.A.2.7 Mapping Positions to Risk Factors
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III.A.2.7.1 Mapping Spot Positions III.A.2.7.2 Mapping Equity Positions III.A.2.7.3 Mapping Zero-Coupon Bonds III.A.2.7.4 Mapping Forward/Futures Positions III.A.2.7.5 Mapping Complex Positions III.A.2.7.6 Mapping Options: Delta and Delta-Gamma Approaches
III.A.2.8 Backtesting VaR Models III.A.2.9 Why Financial Markets Are Not ‘Normal’ III.A.2.10 Summary
III.A.3 Advanced Value at Risk Models Carol Alexander, Elizabeth Sheedy
III.A.3.1 Introduction III.A.3.2 Standard Distributional Assumptions III.A.3.3 Models of Volatility Clustering
III.A.3.3.1 Exponentially Weighted Moving Average (EWMA) III.A.3.3.2 GARCH Models
III.A.3.4 Volatility Clustering and VaR III.A.3.4.1 VaR using EWMA III.A.3.4.2 VaR and GARCH
III.A.3.5 Alternative Solutions to Non-Normality III.A.3.5.1 VaR with the Student’s-t distribution III.A.3.5.2 VaR with EVT III.A.3.5.3 VaR with Normal Mixtures
III.A.3.6 Decomposition of VaR III.A.3.6.1 Stand Alone Capital III.A.3.6.2 Incremental VaR III.A.3.6.3 Marginal Capital
III.A.3.7 Principal Component Analysis III.A.3.7.1 PCA in Action III.A.3.7.2 VaR with PCA
III.A.3.8 Summary
III.A.4.1 Introduction III.A.4.2 Historical Context III.A.4.3 Conceptual Context III.A.4.4 Stress Testing in Practice III.A.4.5 Approaches to Stress Testing: An Overview III.A.4.6 Historical Scenarios
III.A.4.6.1 Choosing Event Periods III.A.4.6.2 Specifying Shock Factors III.A.4.6.3 Missing Shock Factors
III.A.4.7 Hypothetical Scenarios III.A.4.7.1 Modifying the Covariance Matrix III.A.4.7.2 Specifying Factor Shocks (to ‘create’ an event) III.A.4.7.3 Systemic Events and Stress-Testing Liquidity III.A.4.7.4 Sensitivity Analysis III.A.4.7.5 Hybrid Methods
III.A.4.8 Algorithmic Approaches to Stress Testing III.A.4.8.1 Factor-Push Stress Tests III.A.4.8.2 Maximum Loss
III.A.4.9 Extreme-Value Theory as a Stress-Testing Method
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III.B.2.1 Introduction III.B.2.2 What is Default Risk? III.B.2.3 Exposure, Default and Recovery Processes III.B.2.4 The Credit Loss Distribution III.B.2.5 Expected and Unexpected Loss III.B.2.6 Recovery Rates III.B.2.7 Conclusion
III.B.3 Credit Exposure Philipp Schönbucher
III.B.3.1 Introduction III.B.3.2 Pre-settlement versus Settlement Risk
III.B.3.2.1 Pre-settlement Risk III.B.3.2.2 Settlement Risk
III.B.3.3 Exposure Profiles III.B.3.3.1 Exposure Profiles of Standard Debt Obligations III.B.3.3.2 Exposure Profiles of Derivatives
III.B.3.4 Mitigation of Exposures III.B.3.4.1 Netting Agreements III.B.3.4.2 Collateral III.B.3.4.3 Other Counterparty Risk Mitigation Instruments
III.B.4 Default and Credit Migration Philipp Schönbucher
III.B.4.1 Default Probabilities and Term Structures of Default Rates III.B.4.2 Credit Ratings
III.B.4.2.1 Measuring Rating Accuracy III.B.4.3 Agency Ratings
III.B.4.3.1 Methodology III.B.4.3.2 Transition Matrices, Default Probabilities and Credit Migration
III.B.4.4 Credit Scoring and Internal Rating Models III.B.4.4.1 Credit Scoring III.B.4.4.2 Estimation of the Probability of Default III.B.4.4.3 Other Methods to Determine the Probability of Default
III.B.4.5 Market Implied Default Probabilities III.B.4.5.1 Pricing the Calibration Securities III.B.4.5.2 Calculating implied default probabilities
III.B.4.6 Credit rating and credit spreads III.B.4.7 Summary
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III.B.5 Portfolio Models of Credit Loss  Michel Crouhy, Dan Galai, Robert Mark
III.B.5.1 Introduction III.B.5.2 What Actually Drives Credit Risk at the Portfolio Level? III.B.5.3 Credit Migration Framework
III.B.5.3.1 Credit VaR for a Single Bond/Loan III.B.5.3.2 Estimation of Default and Rating Changes Correlations III.B.5.3.3 Credit VaR of a Bond/Loan Portfolio
III.B.5.4 Conditional Transition Probabilities– CreditPortfolioView III.B.5.5 The Contingent Claim Approach to Measuring Credit Risk
III.B.5.5.1 Structural Model of Default Risk: Merton’s (1974) Model III.B.5.5.2 Estimating Credit Risk as a Function of Equity Value
III.B.5.6 The KMV Approach III.B.5.6.1 Estimation of the Asset Value VA and the Volatility of Asset Return III.B.5.6.2 Calculation of the ‘Distance to Default’ III.B.5.6.3 Derivation of the Probabilities of Default from the Distance to Default III.B.5.6.4 EDF as a Predictor of Default
III.B.5.7 The Actuarial Approach III.B.5.8 Summary and Conclusion
III.B.6 Credit Risk Capital Calculation Dan Rosen
III.B.6.1 Introduction III.B.6.2 Economic Credit Capital Calculation
III.B.6.2.1 Economic Capital and the Credit Portfolio Model III.B.6.2.1.1 Time Horizon III.B.6.2.1.2 Credit Loss Definition III.B.6.2.1.3 Quantile of the Loss Distribution
III.B.6.2.2 Expected and Unexpected Losses III.B.6.2.3 Enterprise Credit Capital and Risk Aggregation
III.B.6.3 Regulatory Credit Capital: Basel I III.B.6.3.1 Minimum Credit Capital Requirements under Basel I III.B.6.3.2 Weaknesses of the Basel I Accord for Credit Risk III.B.6.3.3 Regulatory Arbitrage
III.B.6.4 Regulatory Credit Capital: Basel II III.B.6.4.1 Latest Proposal for Minimum Credit Capital requirements III.B.6.4.2 The Standardised Approach in Basel II III.B.6.4.3 Internal Ratings Based Approaches: Introduction III.B.6.4.4 IRB for Corporate, Bank and Sovereign Exposures III.B.6.4.5 IRB for Retail Exposures III.B.6.4.6 IRB for SME Exposures III.B.6.4.7 IRB for Specialised Lending and Equity Exposures III.B.6.4.8 Comments on Pillar II
III.B.6.5 Basel II: Credit Model Estimation and Validation III.B.6.5.1 Methodology for PD Estimation III.B.6.5.2 Point-in-Time and Through-the-Cycle Ratings III.B.6.5.3 Minimum Standards for Quantification and Credit Monitoring Processes III.B.6.5.4 Validation of Estimates
III.B.6.6 Basel II: Securitisation III.B.6.7 Advanced Topics on Economic Credit Capital
III.B.6.7.1 Credit Capital Allocation and Marginal Credit Risk Contributions III.B.6.7.2 Shortcomings of VaR for ECC and Coherent Risk Measures
III.B.6.8 Summary and Conclusions
 
III.C.1.1 Introduction III.C.1.2 Evidence of Operational Failures III.C.1.3 Defining Operational Risk III.C.1.4 Types of Operational Risk III.C.1.5 Aims and Scope of Operational Risk Management III.C.1.6 Key Components of Operational Risk III.C.1.7 Supervisory Guidance on Operational Risk III.C.1.8 Identifying Operational Risk – the Risk Catalogue III.C.1.9 The Operational Risk Assessment Process III.C.1.10 The Operational Risk Control Process III.C.1.11 Some Final Thoughts
III.C.2 Operational Risk Process Models  James Lam
III.C.2.1 Introduction III.C.2.2 The Overall Process III.C.2.3 Specific Tools III.C.2.4 Advanced Models
III.C.2.4.1 Top-down models III.C.2.4.2 Bottom-up models
III.C.2.5 Key Attributes of the ORM Framework III.C.2.6 Integrated Economic Capital Model III.C.2.7 Management Actions III.C.2.8 Risk Transfer III.C.2.9 IT Outsourcing
III.C.2.9.1 Stakeholder Objectives III.C.2.9.2 Key Processes III.C.2.9.3 Performance Monitoring III.C.2.9.4 Risk Mitigation
III.C.3 Operational Value-at-Risk Carol Alexander
III.C.3.1 The ‘Loss Model’ Approach III.C.3.2 The Frequency Distribution III.C.3.3 The Severity Distribution III.C.3.4 The Internal Measurement Approach III.C.3.5 The Loss Distribution Approach III.C.3.6 Aggregating ORC III.C.3.7 Concluding Remarks
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Introduction
If you're reading this, you are seeking to attain a higher standard. Congratulations!
 Those of us who have been a part of financial risk management for the past twenty years, have
seen it change from an on-the-fly profession, with improvisation as a rule, to one with
substantially higher standards, many of which are now documented and expected to be followed.
It’s no longer enough to say  you know. Now, you and your team need to  prove  it.
 As its title implies, this book is the  Handbook for the Professional Risk Manager. It is for those
professionals who seek to demonstrate their skills through certification as a Professional Risk
Manager (PRM) in the field of financial risk management. And it is for those looking simply to
develop their skills through an excellent reference source.
 With contributions from nearly 40 leading authors, the Handbook is designed to provide you
 with the materials needed to gain the knowledge and understanding of the building blocks of
professional financial risk management. Financial risk management is not about avoiding risk.
Rather, it is about understanding and communicating risk, so that risk can be taken more
confidently and in a better way. Whether your specialism is in insurance, banking, energy, asset
management, weather, or one of myriad other industries, this Handbook is your guide.
 We encourage you to work through it sequentially. In Section I, we introduce the foundations of
finance theory, the financial instruments that provide tools for the mitigation or transfer of risk,
and the financial markets in which instruments are traded and capital is raised. After studying this
section, you will have read the materials necessary for passing Exam I of the PRM Certification
program.
In Section II, we take you through the mathematical foundations of risk assessment. While there
are many nuances to the practice of risk management that go beyond the quantitative, it is
essential today for every risk manager to be able to assess risks. The chapters in this section are
accessible to all PRM members, including those without any quantitative skills. The Excel
spreadsheets that accompany the examples are an invaluable aid to understanding the
mathematical and statistical concepts that form the basis of risk assessment. After studying all
these chapters, you will have read the materials necessary for passage of Exam II of the PRM
Certification program.
In Section III, the current and best practices of Market, Credit and Operational risk management
are described. This is where we take the foundations of Sections I and II and apply them to our
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The PRM Handbook
profession in very specific ways. Here the strategic application of risk management to capital
allocation and risk-adjusted performance measurement takes hold. After studying this part, you
 will have read the materials necessary for passage of Exam III of the PRM Certification program.
 Those preparing for the PRM certification will also be preparing for Exam IV - Case Studies,
Standards of Best Practice Conduct and Ethics and PRMIA Governance. This is where we study
some failed practices, standards for the performance of the duties of a Professional Risk
Manager, and the governance structure of our association, the Professional Risk Managers’
International Association. The materials for this exam are freely available on our web site (see
http://www.prmia.org/pdf/Web_based_Resources.htm ) and are thus outside of the Handbook.
 At the end of your progression through these materials, you will find that you have broadened
your knowledge and skills in ways that you might not have imagined. You will have challenged
yourself as well. And, you will be a better risk manager. It is for this reason that we have created
the Professional Risk Managers’ Handbook.
Our deepest appreciation is extended to Prof. Carol Alexander and Prof. Elizabeth Sheedy, both
of PRMIA’s Academic Advisory Council, for their editorial work on this document. The
commitment they have shown to ensuring the highest level of quality and relevance is beyond
description. Our thanks also go to Laura Bianco, President of PRMIA Publications, who has
tirelessly kept the work process moving forward and who has dedicated herself to demanding the
finest quality output. We also thank Richard Leigh, our London-based copyeditor, for his skilful
and timely work.
Finally, we express our thanks to the authors who have shared their insights with us. The
demands for sharing of their expertise are frequent. Yet, they have each taken special time for
this project and have dedicated themselves to making the Handbook and  you  a success. We are
 very proud to bring you such a fine assembly.
Much like PRMIA, the Handbook is a place where the best ideas of the risk profession meet. We
hope that you will take these ideas, put them into practice and certify your knowledge by attaining
the PRM designation. Among our membership are over 300 Chief Risk Officers / Heads of Risk
and 800 other senior executives who will note your achievements. They too know the importance
of setting high standards  and the trust that capital providers and stakeholders have put in them.
Now they put their trust in you and you can prove your commitment and distinction to them.
 We wish you much success during your studies and for your performance in the PRM exams!
David R. Koenig, Executive Director, PRMIA
2004 © The Professional Risk Managers’ International Association xxv
2004 © The Professional Risk Managers’ International Association xxvi
 
Markets
Section I of this Handbook has been written by a group of leading scholars and practitioners and
represents a broad overview of the theory, instruments and markets of finance. This section
corresponds to Exam I in the Professional Risk Manager (PRM) certification programme.
 The modern theory of finance is the solid basis of risk management and thus it naturally
represents the basis of the PRM certification programme. All major areas of finance are involved
in the process of risk management: from the expected utility approach and risk aversion, which
 were the forerunners of the capital asset pricing model (CAPM), to portfolio theory and the risk-
neutral approach to pricing derivatives. All of these great financial theories and their interactions
are presented in Part I.A (Finance Theory). Many examples demonstrate how the concepts are
applied in practical situations.
Part I.B (Financial Instruments) describes a wide variety of financial products and connects them
to the theoretical development in Part I.A. The ability to value all the instruments/assets within a
trading or asset portfolio is fundamental to risk management. This part examines the valuation of
financial instruments and also explains how many of them can be used for risk management.
 The designers of the PRM curriculum have correctly determined that financial risk managers
should have a sound knowledge of financial markets. Market liquidity, the role of intermediaries
and the role of exchanges are all features that vary considerably from one market to the next and
over time. It is crucial that professional risk managers understand how these features vary and
their consequences for the practice of risk management. Part I.C (Financial Markets) describes
 where and how instruments are traded, the features of each type of financial asset or commodity
and the various conventions and rules governing their trade.
 This background is absolutely necessary for professional risk management, and Exam I therefore
represents a significant portion of the whole PRM certification programme. For a practitioner
 who left academic studies several years ago, this part of the Handbook will provide efficient
revision of finance theory, financial instruments and markets, with emphasis on practical
application to risk management. Such a person will find the chapters related to his/her work easy
reading and will have to study other topics more deeply.
 The coverage of financial topics included in Section I of the Handbook is typically deeper and
broader than that of a standard MBA syllabus. But the concepts are well explained and
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appropriately linked together. For example, Chapter I.B.6 on credit derivatives covers many
examples (such as credit-linked notes and credit default swaps) that are not always included in a
standard MBA-level elective course on fixed income. Chapter I.B.9 on simple exotics also
provides examples of path-dependent derivatives beyond the scope of a standard course on
options. All chapters are written for professionals and assume a basic understanding of markets
and their participants.
Finance Theory
Chapter I.A.1 provides a general overview of risk and risk aversion, introduces the utility function
and mean–variance criteria. Various risk-adjusted performance measures are described. A
summary of several widely used utility functions is presented in the appendix.
Chapter I.A.2 provides an introduction to portfolio mathematics, from means and variances of
returns to correlation and portfolio variance. This leads the reader to the efficient frontier,
portfolio theory and the concept of portfolio diversification. Eventually this chapter discusses
normally distributed returns and basic applications for value-at-risk, as well as the probability of
reaching a target or beating a benchmark. This chapter is very useful for anybody with little
experience in applying basic mathematical models in finance.
 The concept of capital allocation is another fundamental notion for risk managers. Chapter I.A.3
describes how capital is allocated between portfolios of risky and riskless assets, depending on
risk preference. Then the efficient frontier, the capital markets line, the Sharpe ratio and the
separation principle are introduced. These concepts lead naturally to a discussion of the CAPM
model and the idea that marginal risk (rather than absolute risk) is the key issue when pricing
risky assets. Chapter I.A.4 provides a rigorous description of the CAPM model, including betas,
systematic risk, alphas and performance measures. Arbitrage pricing theory and multifactor
models are also introduced in this chapter.
Capital structure is an important theoretical concept for risk managers since capital is viewed as
the last defence against extreme, unexpected outcomes. Chapter I.A.5 introduces capital
structure, advantages and costs related to debt financing, various agency costs, various types of
debt and equity, return on equity decomposition, examples of attractive and unattractive debt,
bankruptcy and financial distress costs.
Most valuation problems involve discounting future cash flows, a process that requires
knowledge of the term structure of interest rates. Chapter I.A.6 describes various types of
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interest rates and discounting, defines the term structure of interest rates, introduces forward
rates and explains the three main economic term structure theories.
 These days all risk managers must be well versed in the use and valuation of derivatives. The two
basic types of derivatives are forwards (having a linear payoff) and options (having a non-linear
payoff). All other derivatives can be decomposed to these underlying payoffs or alternatively
they are variations on these basic ideas. Chapter I.A.7 describes valuation methods used for
forward contracts. Discounting is used to value forward contracts with and without intermediate
cash flow. Chapter I.A.8 introduces the principles of option pricing. It starts with definitions of
basic put and call options, put–call parity, binomial models, risk-neutral methods and simple delta
hedging. Then the Black–Scholes–Merton formula is introduced. Finally, implied volatility and
smile effects are briefly described.
Financial Instruments
Having firmly established the theoretical basis for valuation in Part I.A, Part I.B applies these
theories to the most commonly used financial instruments.
Chapter I.B.1 introduces bonds, defines the main types of bonds and describes the market
conventions for major types of treasuries, strips, floaters (floating-rate notes) and inflation-
protected bonds in different countries. Bloomberg screens are used to show how the market
information is presented. Chapter I.B.2 analyses the main types of bonds, describes typical cash
flows and other features of bonds and also gives a brief description of non-conventional
instruments. Examples of discounting, day conventions and accrued interest are provided, as
 well as yield calculations. The connection between yield and price is described, thus naturally
leading the reader to duration, convexity and hedging interest-rate risk.
 While Chapter I.A.7 explained the principles of forward valuation, Chapter I.B.3 examines and
compares futures and forward contracts. Usage of these contracts for hedging and speculation is
discussed. Examples of currency, commodity, bonds and interest-rate contracts are used to
explain the concept and its applications. Mark-to-market, quotation, settlements and other
specifications are described here as well. The principles of forward valuation are next applied to
swap contracts, which may be considered to be bundles of forward contracts. Chapter I.B.4
analyses some of the most popular swap varieties, explaining how they may be priced and used
for managing risk.
 The remaining chapters in Part I.B all apply the principles of option valuation as introduced in
Chapter I.A.8. The power of the option concept is obvious when we see its applications to so
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many instruments and risk management problems. Chapter I.B.5 begins with an analysis of
 vanilla options. Chapter I.B.6 covers one of the newer applications of options: the use of credit
risk derivatives to manage credit risk. Chapter I.B.7 addresses caps, floors and swaptions, which
are the main option strategies used in interest-rate markets. Yet another application of the option
principle is found in Chapter I.B.8 – convertible bonds. These give investors the right to convert
a debt security into equity. Finally, Chapter I.B.9 examines exotic option payoffs. In every case
the author defines the instrument, discusses its pricing and illustrates its use for risk management
purposes.
Financial Markets
Financial risk management takes place in the context of markets and varies depending on the
nature of the market. Chapter I.C.1 is a general introduction to world financial markets. They
can be variously classified – geographically, by type of exchange, by issuers, liquidity and type of
instruments – all are provided here. The importance of liquidity, the distinction between
exchange and over-the-counter markets and the role of intermediaries in their various forms are
explained in more detail.
Money markets are the subject of Chapter I.C.2. These markets are of vital importance to the
risk manager as the closest thing to a ‘risk-free’ asset is found here. This chapter covers all short-
term debt securities, whether issued by governments or corporations. It also explains the repo
markets – markets for borrowing/lending on a secured basis. The market for longer-term debt
securities is discussed in Chapter I.C.3, which classifies bonds by issuer: government, agencies,
corporate and municipal. There is a comparison of bond markets in major countries and a
description of the main intermediaries and their roles. International bond markets are introduced
as well.
Chapter I.C.4 turns to the foreign exchange market – the market with the biggest volume of
trade. Various aspects of this market are explained, such as quotation conventions, types of
brokers, and examples of cross rates. Economic theories of exchange rates are briefly presented
here along with central banks’ policies. Forward rates are introduced together with currency
swaps. Interest-rate parity is explained with several useful examples.
Chapter I.C.5 provides a broad introduction to stock markets. This includes the description and
characteristics of several types of stocks, stock market indices and priorities in the case of
liquidation. Dividends and dividend-based stock valuation methods are described in this
chapter. Primary and secondary markets are distinguished. Market mechanics, including types of
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orders, market participants, margin and short trades, are explained here with various examples
clarifying these transactions. Some exchange-traded options on stocks are introduced as well.
Chapter I.C.6 introduces the futures markets; this includes a comparison of the main exchange-
traded markets, options on futures, specifications of the most popular contracts, the use of
futures for hedging, trade orders for futures contracts, mark-to-market procedures, and various
expiration conventions. A very interesting description of the main market participants concludes
this chapter.
Chapter I.C.7 introduces the structure of the commodities market. It starts with the spot market
and then moves to commodity forwards and futures. Specific features, such as delivery and
settlement methods, are described. The spot–forward pricing relationship is used to decompose
the forward price into spot and carry. Various types of price term structure (such as
backwardation and contango) are described, together with some economic theory. The chapter
also describes short squeezes and regulations. Risk management at the commodity trading desk
is given at a good intuitive level. The chapter concludes with some interesting facts on
distribution of commodity returns.
Finally, Chapter I.C.8 examines one of the most rapidly developing markets for risk – the energy
markets. These markets allow participants to manage the price risks of oil and gas, electricity,
coal and so forth. Some other markets closely linked with energy are also briefly discussed here,
including markets for greenhouse gas emissions, weather derivatives and freight. Energy markets
create enormous challenges and opportunities for risk managers – in part because of the extreme
 volatility of prices that can occur.
 As a whole, Section I gives an overview of the theoretical and practical aspects of finance that are
used in the management of financial risks. Many concepts, some quite complex, are explained in
a relatively simple language and are demonstrated with numerous examples. Studying this part of
the Handbook should refresh your knowledge of financial models, products and markets and
provide the background for risk management applications.
Zvi Wiener, Co-chair of PRMIA’s Education and Standards Committee  
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Measurement
 The role of risk management in financial firms has evolved far beyond the simple insurance of
identified risks. Today it is recognised that risks cannot be properly managed unless they are
quantified. And the assessment of risk requires mathematics. Take, for instance, a large portfolio
of stocks. The relationship between the portfolio returns and the market returns – and indeed
other potential risk factor returns – is typically estimated using a statistical regression analysis.
 And the systematic risk of the portfolio is then determined by a quadratic form, a fundamental
concept in matrix algebra that is based on the covariance matrix of the risk factor returns.
 Volatility is not the only risk metric that financial risk managers need to understand. During the
last decade value-at-risk (VaR) has become the ubiquitous tool for risk capital estimation. To
understand a VaR model, risk managers require knowledge of probability distributions,
simulation methods and a host of other mathematical and statistical techniques. Market VaR is
assessed by mapping portfolios to their risk factors and forecasting the volatilities and
correlations of these factors. The diverse quantitative techniques that are commonly applied in
the assessment of market VaR include eigenvectors and eigenvalues, Taylor expansions and
partial derivatives. Credit VaR can be assessed using firm-value models that are based on the
theory of options, or statistical and/or macro-econometric models. Probability distributions are
even applied to operational risks, though they are very difficult to quantify because the data are
sparse and unreliable. Indeed, the actuarial or loss model approach has been adopted as industry
standard for operational VaR models.
Even if not directly responsible for designing and coding a risk capital model, middle office risk
managers must understand these models sufficiently well to be competent to assess them. And
the risk management role encompasses many other responsibilities. Ten years ago my best
students aspired to become traders because of the high salaries and status – risk management was
 viewed (by some) as a ‘second-rate’ job that did not require very special expertise. Now, this
situation has definitely changed. Today, the middle office risk manager’s responsibility has
expanded to include the independent validation of traders’ models, as well as risk capital
assessment. And the role of risk management in the front office itself has expanded, with the
need to hedge increasingly complex options portfolios. So today, the hallmark of a good risk
manager is not just having the statistical skills required for risk assessment – a comprehensive
knowledge of pricing and hedging financial instruments is equally important.
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No wonder, therefore, that the PRM qualification includes an entire exam on mathematical and
statistical methods. However, we do recognise that many students will not have degrees in
mathematics, physics or other quantitative disciplines. So this section of the Handbook is aimed
at students having no quantitative background at all. It introduces and explains all the
mathematics and statistics that are essential for financial risk management. Every chapter is
presented in a pedagogical manner, with associated Excel spreadsheets explaining the numerous
practical examples. And, for clarity and consistency, we chose two much respected authors of the
highly acclaimed textbook Quantitative Methods in Finance to write the entire section. Keith
Parramore and Terry Watsham have put considerable effort into making the PRM material
accessible to everyone, irrespective of their quantitative background.
 The first chapter, II.A (Foundations), reviews the fundamental mathematical concepts: the
symbols used and the basic rules for arithmetic, equations and inequalities, functions and graphs,
etc. Chapter II.B (Descriptive Statistics) introduces the descriptive statistics that are commonly
used to summarise the historical characteristics of financial data: the sample moments of returns
distributions, ‘downside’ risk statistics, and measures of covariation (e.g. correlation) between two
random variables. Chapter II.C (Calculus) focuses on differentiation and integration, Taylor
expansion and optimisation. Financial applications include calculating the convexity of a bond
portfolio and the estimation of the delta and gamma of an options portfolio. Chapter II.D
(Linear Mathematics and Matrix Algebra) covers matrix operations, special types of matrices and
the laws of matrix algebra, the Cholesky decomposition of a matrix, and eigenvalues and
eigenvectors. Examples of financial applications include: manipulating covariance matrices;
calculating the variance of the returns to a portfolio of assets; hedging a vanilla option position;
and simulating correlated sets of returns. Chapter II.E (Probability Theory) first introduces the
concept of probability and the rules that govern it. Then some common probability distributions
for discrete and continuous random variables are described, along with their expectation and
 variance and various concepts relating to joint distributions, such as covariance and correlation,
and the expected value and variance of a linear combination of random variables. Chapter II.F
(Regression Analysis) covers the simple and multiple regression models, with applications to the
capital asset pricing model and arbitrage pricing theory. The statistical inference section deals
 with both prediction and hypothesis testing, for instance, of the efficient market hypothesis.
Finally, Chapter II.G (Numerical Methods) looks at solving implicit equations (e.g. the Black– 
Scholes formula for implied volatility), lattice methods, finite differences and simulation.
Financial applications include option valuation and estimating the ‘Greeks’ for complex options.
 Whilst the risk management profession is no doubt becoming increasingly quantitative, the
quantification of risk will never be a substitute for good risk management. The primary role of a
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financial risk manager will always be to understand the markets, the mechanisms and the
instruments traded. Mathematics and statistics are only tools, but they are necessary tools. After
 working through this part of the Handbook you will have gained a thorough and complete
grounding in the essential quantitative methods for your profession.
Carol Alexander, Chair of PRMIA’s Academic Advisory Council and co-editor of The PRM Handbook
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Preface to Section III: Risk Management Practices
Section III is the ultimate part of The PRM Handbook in both senses of the word. Not only is it
the final section, but it represents the final aims and objectives of the Handbook. Sections I
(Finance Theory, Financial Instruments and Markets) and II (Mathematical Foundations of Risk
Measurement) laid the necessary foundations for this discussion of risk management practices –
the primary concern of most readers. Here some of the foremost practitioners and academics in
the field provide an up-to-date, rigorous and lucid statement of modern risk management.
 The practice of risk management is evolving at a rapid pace, especially with the impending arrival
of Basel II. Aside from these regulatory pressures, shareholders and other stakeholders
increasingly demand higher standards of risk management and disclosure of risk. In fact, it
 would not be an overstatement to say that risk consciousness is one of the defining features of
modern business. Nowhere is this truer than in the financial services industry. Interest in risk
management is at an unprecedented level as institutions gather data, upgrade their models and
systems, train their staff, review their remuneration systems, adapt their business practices and
scrutinise controls for this new era.
Section III is itself split into three parts which address market risk, credit risk and operational risk
in turn. These three are the main components of risk borne by any organisation, although the
relative importance of the mix varies. For a traditional commercial bank, credit risk has always
been the most significant. It is defined as the risk of default on debt, swap, or other counterparty
instruments. Credit risk may also result from a change in the value of a security, contract or asset
resulting from a change in the counterparty’s creditworthiness. In contrast, market risk refers to
changes in the values of securities, contracts or assets resulting from movements in exchange
rates, interest rates, commodity prices, stock prices, etc. Operational risk, the risk of loss
resulting from inadequate or failed internal processes, people and systems or from external
events, is not, strictly speaking, a financial risk. Operational risks are, however, an inevitable
consequence of any business undertaking. For financial institutions and fund managers, credit
and market risks are taken intentionally with the objective of earning returns, while operational
risks are a by-product to be controlled. While the importance of operational risk management is
increasingly accepted, it will probably never have the same status in the finance industry as credit
and market risk which are the chosen areas of competence.
For non-financial firms, the priorities are reversed. The focus should be on the risks associated
 with the particular business; the production and marketing of the service or product in which
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expertise is held. Market and credit risks are usually of secondary importance as they are a by-
product of the main business agenda.
 The last line of defence against risk is capital, as it ensures that a firm can continue as a going
concern even if substantial and unexpected losses are incurred. Accordingly, one of the major
themes of Section III is how to determine the appropriate size of this capital buffer. How much
capital is enough to withstand unusual losses in each of the three areas of risk? The measurement
of risk has further important implications for risk management as it is increasingly incorporated
into the performance evaluation process. Since resources are allocated and bonuses paid on the
basis of performance measures, it is essential that they be appropriately adjusted for risk. Only
then will appropriate incentives be created for behaviour that is beneficial for shareholders and
other stakeholders. Chapter III.0 explores this fundamental idea at a general level, since it is
relevant for each of the three risk areas that follow.
Market Risk
Chapter III.A.1 introduces the topic of market risk as it is practised by bankers, fund managers
and corporate treasurers. It explains the four major tasks of risk management (identification,
assessment, monitoring and control/mitigation), thus setting the scene for the quantitative
chapters that follow. These days one of the major tasks of risk managers is to measure risk using
 value-at-risk (VaR) models. VaR models for market risk come in many varieties. The more basic
 VaR models are the topic of Chapter III.A.2, while the advanced versions are covered in III.A.3
along with some other advanced topics such as risk decomposition. The main challenge for risk
managers is to model the empirical characteristics observed in the market, especially volatility
clustering. The advanced models are generally more successful in this regard, although the basic
 versions are easier to implement. Realistically, there will never be a perfect VaR model, which is
one of the reasons why stress tests are a popular tool. They can be considered an ad hoc solution
to the problem of model risk. Chapter III.A.4 explains the need for stress tests and how they
might usefully be constructed.
Chapter III.B.1 introduces the sphere of credit risk management. Some fundamental tools for
managing credit risk are explained here, including the use of collateral, credit limits and credit
derivatives. Subsequent chapters on credit risk focus primarily on its modelling. Foundations for
modelling are laid in Chapter III.B.2, which explains the three basic components of a credit loss:
the exposure, the default probability and the recovery rate. The product of these three, which
can be defined as random processes, is the credit loss distribution. Chapter III.B.3 takes a more
detailed look at the exposure amount. While relatively simple to define for standard loans,
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assessment of the exposure amount can present challenges for other credit sensitive instruments
such as derivatives, whose values are a function of market movements. Chapter III.B.4 examines
in detail the default probability and how it can evolve over time. It also discusses the relationship
between credit ratings and credit spreads, and credit scoring models. Chapter III.B.5 tackles one
of the most crucial issues for credit risk modelling: how to model credit risk in a portfolio
context and thereby estimate credit VaR. Since diversification is one of the most important tools
for the management of credit risk, risk measures on a portfolio basis are fundamental. A number
of tools are examined, including the credit migration approach, the contingent claim or structural
approach, and the actuarial approach. Finally, Chapter III.B.6 extends the discussion of credit
 VaR models to examine credit risk capital. It compares both economic capital and regulatory
capital for credit risk as defined under the new Basel Accord.
Operational Risk
 The framework for managing operational risk is first established in Chapter III.C.1. After
defining operational risk, it explains how it may be identified, assessed and controlled. Chapter
III.C.2 builds on this with a discussion of operational risk process models. By better
understanding business processes we can find the sources of risk and often take steps to re-
engineer these processes for greater efficiency and lower risk. One of the most perplexing issues
for risk managers is to determine appropriate capital buffers for operational risks. Operational
 VaR is the subject of Chapter III.C.3, including discussion of loss models, standard functional
forms, both analytical and simulation methods, and the aggregation of operational risk over all
business lines and event types.
 
I.A.1 Risk and Risk Aversion
 Jacques Pézier1
I.A.1.1 Introduction
Risk management, in a wide sense, is the art of making decisions in an uncertain world. Such
decisions involve a weighting of risks and rewards, a choice between doing the safe thing and
taking a risk. For example, we may ponder whether to invest in a new venture, whether to
diversify or hedge a portfolio of assets, or at what price it would be worth insuring a person or a
system. Risk attitude determines such decisions. Utility theory offers a rational method for
expressing risk attitude and should therefore be regarded as a main pillar of risk management.
 The other two pillars of risk management are the generation of good alternatives – without which
there would be nothing to decide – and the assessment of probabilities – without which we could
not tell the likely consequences of our actions.
Rationality, in the context of utility theory, means simply that decisions should be logically 
consistent with a set of preference axioms and in line with patterns of risk attitude expressed in
simple, easily understood circumstances. So, utility theory does not dictate what risk attitude
should be – that remains a personal matter or a matter of company policy – it merely provides a
logical framework to extend risk preferences from simple cases to complex situations.
But why should one seek an axiomatic framework to express risk preferences? Alas, experience
shows that unaided intuition is an unreliable guide. It is relatively easy to construct simple
decision problems where intuitive choices seem to contradict each other, that is, seem to violate
basic rules of behaviour that we hold as self-evident. It seems wise, therefore, to start by 
agreeing a basic set of rules and then draw the logical consequences.
 Thus, utility theory is neither purely descriptive nor purely normative. It brings about a more
disciplined, quantitative approach to the expression of risk attitude than is commonly found in
everyday life. In other words, where too often risk taking is ‘seat of the pants’ or based on ‘gut
feel’ or ‘nose’, it tries to bring the brain into play. By questioning instinctive reactions to risky 
situations, it leads decision makers and firms to understand better what risk attitude they ought to
adopt, to express it formally as an element of corporate policy and to convey it through the
organisation so that decisions under uncertainty can be safely delegated.
1 Visiting Professor, ISMA Centre, University of Reading, UK.
 
The PRM Handbook -– I.A.1 Risk and Risk Aversion 
 This chapter introduces some concepts that are absolutely fundamental to the management of 
financial risks. Section I.A.1.2 introduces the idea of utility maximisation following Bernoulli’s
original ideas. Section I.A.1.3 discusses the ‘axiom of independence of choice’, one of the basic
axioms that must be satisfied if preferences over risky outcomes are to be represented by a utility 
function. Section I.A.1.4 introduces the principle of maximum expected utility and the concept of 
risk aversion (and its inverse, risk tolerance). Section I.A.1.5 explains how to encode your
personal attitude to risk in your own utility function. Section I.A.1.6 shows under what
circumstances the principle of maximum expected utility reduces to a mean–variance criterion to
distinguish between different investments. A comprehensive treatment of risk-adjusted
performance measures is given in Section I.A.1.7. We pay particular attention to the
circumstances in which the risks to be compared are not normally distributed and investors are
mainly concerned with downside risks. Section I.A.1.8 summarises and indicates which types of
decision criteria and performance measures may be appropriate in which circumstances.
Much of the material that is introduced in this chapter will be more fully discussed in other parts
of the Handbook. Thus you will find many references to subsequent chapters in Part I.A, Part II
and Part III of the Handbook. A thorough treatment of utility theory, whilst fundamental to our
understanding of risk and risk aversion , is beyond the scope of the PRM exam. However, for
completeness, and for readers seeking to use this chapter as a resource that goes further than the
PRM syllabus, we have provided extensive footnotes of the mathematical derivations.
Furthermore, we have added an Appendix that describes the properties of standard utility
functions. However, it should be stressed that neither the mathematical derivations in the
footnotes nor the material in Appendix B are part of the PRM exam.
I.A.1.2 Mathematical Expectations: Prices or Utilities?
It may seem curious nowadays that early probabilists, who liked to study games of chance, took it
for granted that the mathematical expectation of cash outcomes was the only rational criterion
for choosing between gambles. The expected value  of a gamble is defined as the sum of its cash
outcomes weighted by their respective probabilities; the gamble with the highest expected value
 was deemed to be the best. Fairness in gambling was the main argument in support of this
principle (among ‘zero-sum’ games, where the gains of one player are the losses of the other, only 
zero-expectation games are fair). Another argument drew on the weak law of large numbers, which
implies that, if the consequences of each gamble are small relative to the wealth of the players,
then, in the long run, after many independent gambles, only the average result would matter.
Daniel Bernoulli (1738) was the first mathematician to question the principle of maximising 
expected value and to try to justify departures from it observed in daily life. He questioned
 
The PRM Handbook -– I.A.1 Risk and Risk Aversion 
choices that fly in the face of the principle of maximising expected value. For example, he asked,
if a poor man were offered an equal chance to win a fortune or nothing, should he be regarded as
irrational if he tried to negotiate a sure reward of slightly less than half the potential fortune? Or
is it insane to insure a precious asset and thus knowingly contribute an expected profit to the
insurance company and therefore an equivalent expected decrease in one’s wealth? To reconcile
common behaviour with a maximum-expectation principle, Bernoulli suggested applying the
principle not to cash outcomes but to utilities2  associated with cash outcomes. Bernoulli thus
pre-dates by half a century the core tenet of the Utilitarianism school of social philosophy, the
distinction between:
and
the price , i.e. the ‘exchange value’ of an asset.
Bernoulli’s principle was that actions should be directed at maximising expected utility . The problem
that inspired Bernoulli and which has gained fame under the name of the St Petersburg paradox 3
runs as follows: Peter tosses a coin and continues to do so until it should land ‘heads’. He agrees
to give Paul one ducat if he gets ‘heads’ on the very first throw, two ducats if he gets it on the
second, four if on the third, eight if on the fourth, and so on, so that with each additional throw 
the number of ducats he must pay is doubled. We seek to determine the value of Paul’s
expectation.
Since, with a fairly tossed, symmetrical coin, the probability of landing heads for the first time on
the kth toss is 2 – k  and the corresponding reward is 2k  – 1 ducats, the contribution to Paul’s
monetary expectation of this outcome is half a ducat. And since there is an infinite number of 
possible outcomes k = 1, k = 2, etc., Paul’s monetary expectation is infinite. But, then as now,
gamblers are not willing to pay more than a few ducats for the right to play the game, hence the
paradox.
Bernoulli suggested that the utility of a cash reward depends on the existing wealth of the
recipient. He even made the far stronger assumption that utility is always inversely proportional to
existing wealth , in other words, that a gain of one  ducat to someone worth a thousand ducats has
the same utility as a gain of a thousand  ducats to someone worth a million ducats.
In this case a small change in utility, du , would be related to a small change in wealth, dx , by 
du  = dx /x .
2 In the Latin original, to calculate an ‘emolumentum medium’. 3 Simply because Bernoulli’s paper was published in the Commentaries from the Academy of Sciences of St Petersburg .
 
The PRM Handbook -– I.A.1 Risk and Risk Aversion 
 This leads, by integration (see Section II.C.6), to a logarithmic utility function,
u ( x  ) = ln ( x  ).
If we apply a logarithmic utility to the St Petersburg paradox, it no longer appears to be a
paradox. For instance, a gambler whose only wealth is the game itself would perceive an expected
utility of ln(2), which is the same as the utility of 2 ducats, and this is quite a small number – far
short of infinity! In general, if the gambler has a logarithmic utility function, the larger the initial
 wealth of the gambler, the larger his perceived utility of the game.
I.A.1.3 The Axiom of Independence of Choice
Rarely is the power of a new idea fully understood on first encounter. Bernoulli’s introduction of 
a utility function did   influence the development of classical economics, where it was transposed
into a deterministic context. But it took more than two hundred years for the concept to be
revived in its original probabilistic context and to be re-erected on a firmer footing. In a seminal
book on games theory the mathematician J. von Neumann and the economist O. Morgenstern
(1947) postulated a basic set of rules from which it will follow that a utility function provides a
complete  description of an individual’s risk attitude.4
Bernoulli made a very strong assumption – that the utility of a ga